[LON-CAPA-users] formula response and maxima floating point numbers

[Peter Riegler]

Hi Lon!

On 09.09.2010 15:54, Lon H Mitchell wrote:
> According to the Maxima manual, a shortcut to getting a floating point
> approximation to a function value is to use a decimal argument. For
> example, sqrt(2.0) will return 1.414213562373095 while sqrt(2) will stay
> as sqrt(2).
> This shortcut can result in some seemingly incomprehensible answers. For
> example, "is(sqrt(1.5) = sqrt(3/2))" returns /false/.

Note that "is" checks on syntactical equality (see documentation of "is" 
in maxima, e.g. via "? is;" without double quotes on a maxima prompt). 
Contrast this with "equal" which tests on equivalence. See the manual 
for the exact meaning of equivalence (roughly ratsimp(lhs-rhs)).
What formularesponse does, is testing for equivalence in a more general 
sense, namley trigsimp(trigreduce(lhs-rhs)).

> Further, a formula
> response problem with answer "1+ sqrt(3/2)*x" will mark "1 +
> sqrt(1.5)*x" as incorrect and vice versa. Since the behavior is inherent
> to Maxima, math response problems can be similarly affected.
Actually this is inherent to computer algebra systems. It is a question 
of precision. For instance 1 or 1/3 are expressions that have infinite 
precision while 1. or 0.333 have finite precision (and are of different 
type, namely float).
> Note that this behavior only seems to arise when dealing with a function
> value. For example, "x^(1.5)" and "x^(3/2)" are considered equivalent by
> Maxima, as are "1.5" and "3/2".
>
> While one possible solution is to use sampling rather than algebraic
> checking, my guess would be that sqrt(1.5) = sqrt(3/2) would be an
> expected equivalence on the part of most users.
Sadly, that is true. Even Mathematica seems to have given up on that 
issue. Up to version 4 Mathematica strictly (an rightly - in my opinion) 
differentiated between e.g. 1 and 1.0. Now equality is defined somewhat 
arbitrarily. The manual says "Approximate numbers are considered equal 
if they differ in at most their last eight binary digits (roughly their 
last two decimal digits)."

I couldn't find any documentation on how maxima precisely is handling 
the issue. However, it seems to me that maxima converts floats to 
rationals, e.g.
(%i28) is(equal(3/2,1.5));
`rat' replaced 0.0 by 0/1 = 0.0
(%o28)                               true
(%i29) is(equal(0.333,1/3));
`rat' replaced -3.333333333332966E-4 by -1/3000 = -3.333333333333333E-4
(%o29)                               false
(%i30) is(equal(1/7,0.142857));
`rat' replaced 1.4285714283746032E-7 by 1/7000000 = 1.4285714285714285E-7
(%o30)                               false

>
> The question then: is there a way to instruct Maxima to turn this
> shortcut behavior off,
I don't know of any environment variable or flag that forces maxima to 
do so. I even doubt that there is such a thing.

> and, if so, is it possible to change formula
> response problems (or lonmaxima) to make use of it?
To do so we would need to precisely understand what maxima is doing. And 
we would need to be confident that maxima won't change that behavior 
(cf. Mathematica).

My first recommendation would be to use mathresponse and to check the 
correctness via something like
is(equal(float(1+ sqrt(3/2)*x),float(1+ sqrt(1.5)*x)));
However, I am not quite sure what this is doing, as I don't exactly know 
how maxima is handling equality of floats and rationals (see above).

Note that the whole issue is very similar to significant digits. Most 
users would expect 0.5 and 0.500 to be equivalent. Some, in particular 
physics authors, want loncapa to behave "scientific". The way to handle 
this discrepancy between "most users" and "scientific users" is to have 
a parameter by which you could switch of significant digits. If we can 
handle the issue maxima-wise we would need to have such a parameter 
instead of changing the current (limited) code of lonmaxima.

I hope this helps to at least understand the trickiness of the issue.

Maybe you could contact that maxima people to find out more on this.

Peter


>
> Thanks,
>
> Lon Mitchell
>
>
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-- 
Peter Riegler
Fakultät Informatik
Ostfalia Hochschule für angewandte Wissenschaften
Fachhochschule Braunschweig/Wolfenbüttel
Salzdahlumer Str. 46/48
38302 Wolfenbüttel
Fon: ++49 5331 939 31540
http://public.rz.fh-wolfenbuettel.de/~riegler